Mathematics musings: the Common Core, algebra, and instruction

It started when a friend tweeted this article about a charter school in San Diego that had revamped its math curriculum so that their kids would be better prepared for algebra. The school found that its students “were developing too many shortcuts and not enough understanding.” They were procedurally compliant, it seems, but were not developing deep conceptual knowledge. When these previously mathematics whiz kids hit algebra, their test “scores plummeted.”

The article shared this graphic from the San Diego school district:

For more on this procedural/conceptual dynamic duo, see this transcript and video from Harvard’s Jon Star. (Full disclosure: Jon and I co-wrote this piece for The Center for Comprehensive School Reform and Improvement.) Jon states that “procedural knowledge and conceptual knowledge develop iteratively, that they sort of feed into one another…It’s not so much that if you only do procedures then the concepts sort of develop by themselves or if you only do concepts for a long time then when you get around to the procedures, they’re super easy. We haven’t found that to be the case.”

It was interesting that the San Diego school saw scores fall when kids hit algebra, given the importance of this subject matter. The final report of the National Mathematics Advisory Panel stated that algebra “is a demonstrable gateway to later achievement. Students need it for any form of higher mathematics later in high school; moreover, research shows that completion of Algebra II correlates significantly with success in college and earnings from employment. In fact, students who complete Algebra II are more than twice as likely to graduate from college compared to students with less mathematical preparation.”

Unfortunately, mathematics teachers generally feel that students are not adequately prepared to take Algebra I, as shown by this report on the National Survey of Algebra Teachers, with data collected from 743 of them in the spring of 2007.

My circuitous math path continued. There was the article on the San Diego school, and then the other night at my son’s school’s PTA meeting, there was a quick report on the just finished work and recommendations of our district’s K-12 Mathematics Joint Work Group, which had been charged with exploring “the complex issues surrounding mathematics teaching and learning in” the district and developing “recommendations on ways to improve student achievement in mathematics system-wide.” The group was formed about a year ago and ultimately devised this set of recommendations, after 18 meetings:

I’m unsure what some of these mean – “Aligning school structures and strategies”? – but I was interested to read about “students skipping grades” and the need to “refocus targets,” and more is said about these issues in this Washington Post article, posted at this blog. The article states that these new changes come “as high school teachers were increasingly saying that even their advanced students were arriving in class unprepared…School officials said more than half of fifth-graders are taking sixth-grade math or higher.” A district deputy supe stated that it “was better to tackle topics in greater depth.”

Alignment to the Common Core will drive more in-depth study. See this slide from the PowerPoint deck that the math work group used for its final presentation.

As I compare the two, I’m struck by the coverage (I use that word pejoratively) that happens now with our district’s curriculum – that, for example, algebra study spans all grade levels, with little intensity of focus at any one – while the Common Core demands intense study. Look at 8th grade, as students dive deeply into mostly algebra and geometry. No more mile wide, inch deep.

What interests me a great deal, when I think about these new standards and their new approach to subject matter, is teacher training. Many states that have adopted the Common Core feel the same way. According to a recent report from the Center on Education Policy, 44 states have fully or provisionally adopted these common language arts and math standards, 33 plan to make changes to professional development for teachers, and 21 of 33 plan to do it by 2012 or earlier, an ambitious undertaking.

Let me see if I can circle back to my start: The kids in that San Diego school became procedurally compliant because they were taught that way, and to make changes in that school district, “thousands of teachers have been trained in the new [teaching] methods.” These new methods evolve from, as the National Mathematics Advisory Panel stated, “a more advanced perspective [of] the mathematical content [teachers] are responsible for teaching and the connections of that content to other important mathematics.” Denise Mewborn, Professor of Mathematics Education at the University of Georgia, stated in another piece I wrote for The Center that teachers “need to learn to solve a problem several different ways, compare and contrast the various solution strategies, explain the connections among the strategies, explain why each strategy works, and consider things such as which strategies they would highlight in a classroom situation, for what purpose, and in what order.”

All of this suggests a significant facility with mathematics, an expertise that comes with deep knowledge. Yes, new texts and materials that are aligned with the Common Core will be critical to student success. But even more important will be the ongoing training that teachers get, a mix of content and methods. Elementary school teachers will be the busiest. They are critical to the success of this Common Core initiative yet have more than mathematics to teach. States and districts must think carefully about professional development for them, to ensure that they are prepared for this change. Will they be ready to, as Denise Mewborn wrote, “solve a problem several different ways, compare and contrast the various solution strategies, explain the connections among the strategies,” and “explain why each strategy works”? Whoa: That is a lot. But that sounds to me like a darn interesting and valuable class – and a class that will prepare students for thoughtful and successful mathematics study in the future.